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Integrability for the Jacobian of orientation preserving mappings. (English) Zbl 0847.26012
The following theorem is proved: Let \(\Omega\subset \mathbb{R}^n\) be a bounded domain, let \(f: \Omega\to \mathbb{R}^n\), \(n\geq 2\), and let \(Jf(x)\) be the Jacobian of the mapping \(f\), i.e., \(Jf(x):= \text{det } Df(x)\). Assume that \(Jf\geq 0\) and \(f\in L^n (\log L)^{- s}(\Omega)\), i.e., \(\int_\Omega |Df(x)|^n[\log(1+ |Df(x)|)]^{- s} dx< \infty\), \(0\leq s\leq 1\); then \(Jf\in L^1(\log L)^{1- s}_{\text{loc}}\), i.e., \[ \int_K Jf(x)[\log(1+ Jf(x))]^{1-s} dx< \infty \] for any compact subset \(K\) of \(\Omega\). The theorem fills the gap between S. Müller’s result [J. Reine Angew. Math. 412, 20-34 (1990; Zbl 0713.49004)] obtained for \(s= 0\) and a result of T. Iwaniec and C. Sbordone [Arch. Ration. Mech. Anal. 119, No. 2, 129-143 (1992; Zbl 0766.46016)] obtained for \(s= 1\). Moreover, it is shown that the conjecture \(|Df|\in\) Lorentz space \(L^{n, q}(\Omega)\), \((n< q< \infty)\Rightarrow Jf\in L(\log L)^{n/q}_{\text{loc}}(\Omega)\) is false.

MSC:
26B10 Implicit function theorems, Jacobians, transformations with several variables
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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